From Joe Hickerson's column in Sing Out!, it's clear that any song search is gong to have to look for "Bartimus" "Bartemas" and "Barnabus"

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From http://www.naxosdirect.ca/GOLDEN-GATE-QUARTET-Gospel-Train-1937-1942/title/8120731/
Blind Barnabas (more info)
Composed by: Willie Johnson [one of the members of the Golden Gate Quartet]
Recording date: 25 May 1941
Produced by: Lennick, David

Blind Barnabas - 02:11:

Clicking "Read more about this recording" yields a lot more information about Willie Johnson. Willie Johnson can certainly be credited with the Golden Gate Quartet's arrangement, but is the actual composer?

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From http://www.allmusicguide.com/cg/amg.dll?p=amg&sql=11:3ifoxqq5ldje~T3, a listing that calls it "Traditional":

[song title:] Blind Barnabas         [composer:] Traditional         [on the album:] Warrior on the Battlefield: A Capella Trail Blazers -- 1920s-1940s

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Sorry to present resources with no conclusion, but the next researcher can refine these bits of information further.

--Charlie Baum

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Golden Gate Quartet performance (fragment): https://www.youtube.com/watch?v=i5K0aFP8mSw

Full performance by Golden Gate Quartet: https://www.youtube.com/watch?v=wG0uIRJvEho